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Let Γ = (G , σ) be a signed graph. The spectral radius of Γ is the largest absolute value of its adjacency eigenvalues. In this paper we identify the real numbers which are limit points of spectral radii of signed graphs. This is one of the two aspects of a problem in spectral graph theory known as the Hoffman program, implemented here for signed graphs. [ABSTRACT FROM AUTHOR]

• Stable and homogeneous of droplets for the aqueous two-phase system is achieved. • Neck evolution of water-in-water droplets in flow-focusing devices is self-similar. • Four flow patterns were observed and analyzed. • Formation process of droplet includes slow and fast squeezing, and pinch-off stages. • Evolution regulations of droplet neck interface in three stages were described. In ATPS, the ultra-low interfacial tension makes the stable generation of droplets very difficult. In this paper, the stable and homogeneous preparation of ATPS droplets is achieved over a wide range of flow rates, and the self-similarity analysis and formation dynamics of the droplets are studied. Formation of ATPS droplets show good self-similarity. The generation process of ATPS droplets can be divided into slow squeezing, fast squeezing and rapid pinch-off stages. In the slow squeezing stage, the dimensionless minimum neck width w m / w c is linear with the time, and the absolute value of the slope increases with the capillary number of continuous phase Ca c. In the rapid squeezing and rapid pinch-off stages, the w m / w c satisfies the power-law relationship with time, and the absolute value of the power-law indexes decrease with increasing Ca c , and the pre-exponential factor of pinch-off stage increases with increasing Ca c. [ABSTRACT FROM AUTHOR]